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Abstract <p>For every orthonormal system of functions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\{\varphi_{n}\}\)</EquationSource> <!--ContMath2570020Dilanyan-m1--> </InlineEquation> on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((0,\,1)\)</EquationSource> <!--ContMath2570020Dilanyan-m2--> </InlineEquation>, pointwise bounded by a function <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(f\in L^{2}(0,\,1)\)</EquationSource> <!--ContMath2570020Dilanyan-m3--> </InlineEquation>, that is, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(|\varphi_{n}(x)|\leq f(x)\)</EquationSource> <!--ContMath2570020Dilanyan-m4--> </InlineEquation>, it is possible to construct an orthonormal system <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\{\psi_{n}\}\)</EquationSource> <!--ContMath2570020Dilanyan-m5--> </InlineEquation> with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(|\psi_{n}(x)|\equiv 1\)</EquationSource> <!--ContMath2570020Dilanyan-m6--> </InlineEquation>, such that the series <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\sum a_{n}\varphi_{n}\)</EquationSource> <!--ContMath2570020Dilanyan-m7--> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\sum a_{n}\psi_{n}\)</EquationSource> <!--ContMath2570020Dilanyan-m8--> </InlineEquation> converge almost everywhere simultaneously, for every <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\{a_{n}\}\in\ell^{2}\)</EquationSource> <!--ContMath2570020Dilanyan-m9--> </InlineEquation>.</p>

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Equivalence between Some Orthonormal Systems

  • 期刊论文

  • 发表日期：

    2026年2月1日

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  • V. G. Dilanyan

摘要

Abstract

For every orthonormal system of functions \(\{\varphi_{n}\}\) on \((0,\,1)\) , pointwise bounded by a function \(f\in L^{2}(0,\,1)\) , that is, \(|\varphi_{n}(x)|\leq f(x)\) , it is possible to construct an orthonormal system \(\{\psi_{n}\}\) with \(|\psi_{n}(x)|\equiv 1\) , such that the series \(\sum a_{n}\varphi_{n}\) and \(\sum a_{n}\psi_{n}\) converge almost everywhere simultaneously, for every \(\{a_{n}\}\in\ell^{2}\) .